The dynamical connection between the stratosphere and the troposphere in the Northern Hemisphere winter is investigated with general circulation model (GCM) simulations under perpetual January conditions. The variability in the stratosphere is strongly dominated by vacillations on a timescale of 100 days. One-point correlation maps of the zonal mean zonal wind reveal the characteristic downward propagation of the stratospheric disturbances. The meridional structure of the stratospheric vacillations is well described by a few empirical orthogonal functions (EOFs) of geopotential height. The leading EOF describes a standing oscillation of the stratosphere, while the two next describe the vertical and horizontal propagation. In the troposphere the leading EOF of the surface pressure shows the characteristic circumpolar structure of the Arctic Oscillation. The covariance between the leading EOF of the surface pressure and the zonal mean zonal wind reveals a vertical structure similar to the observed below 10 hPa, while the sign of the covariance changes above 10 hPa where observations are scarce. The downward propagating stratospheric modes play a prominent role in the vertical coupling. Singular value decomposition of the cross-covariance matrix between the stratospheric zonal mean zonal wind and the surface pressure shows that a significant part of the tropospheric variability can be related to the downward propagating anomalies. The leading pattern of surface pressure anomalies found in this way closely resembles the pattern of the Arctic Oscillation and describes 20--30% of the total variance. This result is confirmed by studying the covariance between the geopotential height fields and the stratospheric principal components. ¿ 2000 American Geophysical Union